Monday, June 23, 2008

Quine on Grammatical Structure and Logical Truth

In Philosophy of Logic, Quine offers the following definition of "logical truth": "a logical truth is a truth that cannot be turned false by substituting for lexicon. When for its lexical elements we substitute any other strings belonging to the same grammatical categories, the sentence is true." (2nd ed., 58)

Later on, considering whether to strengthen FOL to allow for adverbial modification of predicates, Quine claims that, on the definition of "logical truth" lately quoted:

the sentence

(5) ~(Ex)(x walks rapidly . ~(x walks)),

or 'Whatever walks rapidly walks', would qualify as logically true. (76)

This is interesting. The grammatical definition of logical truth is both epistemologically interesting and clears up a lot of confusions I have about the relationship between formal logic and natural language. I still don't know whether it adequately captures all of the intuitive cases of logical truth.

Really, my only observation here is that it seems that the grammatical definition does not make (5) a logical truth. This is because adverbs can sometimes alienate the predicates they modify. An adverb A alienates a predicate F in a sentence token S iff removal of A from S would change the truth-value of the clause of which F is a part. Briefly, A alienates F (in a certain context) if something can be F A'ly without being F simpliciter. Consider the following cases of adverbial alienation:

(1) Tim indirectly told John about Sally.
(2) Paul is coming home shortly.
(3) Sue allegedly stole the watch.
(4) Esther nearly won the tennis match.

We can imagine cases in which (1), (2), (3), and (4) are true, but their non-adverbialized counterparts aren't. There doesn't seem to be anything syntactically unusual about these adverbs. By all appearances, (1), (2), (3), and (4) have the same grammatical structure, respectively, as the following:

(1`) Tim excitedly told John about Sally.
(2`) Paul is coming home currently.
(3`) Sue actually stole the watch.
(4`) Esther barely won the tennis match.

But if (1), (2), (3), and (4) can be turned from truth to falsehood by transformation into (1`), (2`), (3`), and (4`) then, by the grammatical definition, none of these are logically true.

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